G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Recursive Triangulation UsingBearings-Only Sensors
G. Hendeby, LiU, Sweden
R. Karlsson, LiU, Sweden
F. Gustafsson, LiU, Sweden
N. Gordon, DSTO, Australia
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Motivating Problem
Known to be difficult to estimate
Highly nonlinear, especially at short range
Previously used to demonstrate usefulness of new methods
Methods and performance measures will be discussed
Track a target during close fly-by using bearings only sensors
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Filters
The following filters have been evaluated and compared
Local approximation: Extended Kalman Filter (EKF) Iterated Extended Kalman Filter (IEKF) Unscented Kalman Filter (UKF)
Global approximation: Particle Filter (PF)
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Filters: (I)EKF
EKF: Linearize the model around the best estimate and apply the Kalman filter (KF) to the resulting system.
IEKF: Relinearize the model after a measurement update with a (hopefully) improved estimate, and restart the update with this linear model.
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Filters: UKF
Simulate carefully chosen “sigma points” to transform involved covariance matrices and use in the KF.
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Filters: PF
Simulate several possible states and compare to the measurements obtained.
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Filter Evaluation
Root mean square error (RMSE) Standard performance measure Bounded by the Cramér-Rao Lower Bound (CRLB) Ignores higher order moments
Kullback divergence Compares the distance between two distributions Captures effects not seen in the RMSE
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Test Setup
Measurements from: Initial estimate: Initial estimate covariance: Different target positions along the -axis have been
evaluated. Poor initial information
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Test Setup: Measurement Noise
Gaussian noise:
Gaussian mixture noise:
Generalized Gaussian noise:
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Test Setup: True Inferred Distribution
True inferred state distribution for one noise realization,
Some non-Gaussian features
Computed using gridding, not feasible for use in practice
CRLB for this situation:
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Comparison: RMSE
The PF is overall best, however CRLB is not reached
(I)EKF sometimes diverges, iterating then could be catastrophic
Difficult to extract information from non-Gaussian measurements
Higher moments are ignored in this comparison
Gaussian mixture noise
Generalized Gaussian noise
50 measurements
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Comparison: Kullback divergence
The Kullback divergence has been used to capture other differences between estimated and true distribution. Note, the results represents only one realization.
Here: Gaussian mixture noise and
Filter No. measurements0 1 2 3 4 5
EKF 3.16 10.15 10.64 11.53 10.81 11.23
IEKF 3.16 10.12 10.40 11.55 11.14 11.61
UKF 3.16 10.15 10.62 11.53 11.14 11.63
PF 3.32 9.17 8.99 8.87 9.87 9.98
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Conclusions
A bearings-only estimation problem, with large initial uncertainty, has been studied using different filters.
As a complement to comparing RMSE, the Kullback divergence has been used to capture more than the variance aspects of the obtained estimates.
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham
Conclusions, cont’d
(Iterated) Extended Kalman Filter – ((I)EKF) Works acceptable with good initial information, but has
difficulties with bad initial information Iterating often slightly improve performance, but sometimes
backfires badly
Unscented Kalman Filter (UKF) Results are not bad, but not as impressive as suggested in
recent literature
Particle Filter (PF) Works well at the price of higher computational effort
G. HendebyRecursive Triangulation Using Bearings-Only Sensors
TARGET ‘06Austin Court, Birmingham